Building on the hint of the time-ordering through the temporal \(\theta \) -functions in the Ward identities, we formulate the operator formalism for \(1+1\) D CCFTs. We show that there is no need to perform a `radial quantization' by assuming a `plane-to-cylinder' map [1]}. Rather it is the magic of the temporal \(\theta \) -function that executes a vital role in relating the operator product expansions (OPEs) with the operator commutation relations via a contour integral over the complex variable \(x\) . This relation has been implicitly used in the literature, e.g. in [2]}, as it is what is expected from the `limiting' procedure. But, to the best of our knowledge, an intrinsic derivation using only the principles of \(1+1\) D Carrollian physics has not appeared before.
